SYMMETRY ANALYSIS AND SOME SOLUTIONS OF GOWDY EQUATIONS

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1 SYMMETRY ANALYSIS AND SOME SOLUTIONS OF GOWDY EQUATIONS RAJEEV KUMAR 1, R.K.GUPTA 2,a, S.S.BHATIA 2 1 Department of Mathematics Maharishi Markandeshwar Univesity, Mullana, Ambala Haryana, India rajeevkumarbudhiraja@gmail.com. 2 School of Mathematics and Computer Applications Thapar University, Patiala Punjab, India a : rajeshgupta@thapar.edu Corresponding author Received April 11, 2014 The Gowdy spacetime metric splits the vacuum Einstein field equations into evolution equations, known as Gowdy equations. In this paper, Lie classical method is applied to investigate symmetries of the Gowdy equations, that is a coupled system of nonlinear second order partial differential equations. Symmetries which are obtained in this paper, are further helpful for reducing the coupled system of partial differential equations into ordinary differential equations. Moreover, some new solutions of Gowdy equations are derived from reduced differential equations. Key words: Lie classical method, Gowdy spacetimes, Gowdy equations, Cosmology. PACS: Sv, k. 1. INTRODUCTION In the literature, different approaches have been applied to study the singularity in inhomogeneous cosmologies. The most extensively studied class of these spacetimes is the class of Gowdy spacetimes on T 3 R [1]-[4]. In the Gowdy class the spacetime metric can be written in the form ds 2 = t 1 2 e λ 2 dt 2 + dθ 2 + t e P dx + Qdy 2 + e P dy 2}. 1 Here the functions P, Q and λ depend only on t and θ. The essential equations describing these spacetimes are a system of semilinear wave equations for two functions P and Q which are assumed to be periodic with period 2π in the spatial coordinate t. The vacuum Einstein field equations [5] split into two systems, in which first is evolution equations for P and Q: P tt + 1 t P t P θθ e 2P Q 2 t Q 2 θ = 0 Q tt + 1 t Q t Q θθ + 2Q t P t Q θ P θ = 0, RJP Rom. 60Nos. Journ. Phys., 1-2, Vol , Nos , P. c 15 21, 2015 Bucharest, - v.1.3a*

2 16 Rajeev Kumar, R.K.Gupta, S.S.Bhatia 2 and the other is constraint equations for λ: λ θ = t [ P 2 θ + P 2 t + e 2P Q 2 t + Q 2 θ], λ t = 2t P t P θ + e 2P Q t Q θ. 3 The constraint equations determine λ once P and Q are known. The integrability conditions for the constraint equations are satisfied as a consequence of the evolution equations. Since the evolution equations do not depend on λ there is essentially a complete decoupling of constraints from evolution equations. In this paper, we will deal with equations 2, which we call the Gowdy equations [6, 7]. The layout of this paper is as follows: Section 2 is devoted to an outline of the Lie classical method [8] [10] for generating various symmetries of the Gowdy equations. Section 3 deals with the determination of the transformation group for the reduction of system 2 to a system of ordinary differential equations ODEs and exact solutions have also been obtained. We record our conclusion in last Section. 2. LIE SYMMETRY ANALYSIS A Lie point symmetry of a partial differential equation PDE is an invertible transformation of the dependent and independent variables that leaves the equation unchanged. Sophus Lie observed that if we restrict ourself to symmetries that depend continuously on a small parameter and that form a group continuous one-parameter group of transformations, one can linearize the symmetry conditions and end up with an algorithm for calculating continuous symmetries. The Lie classical method has been used to examine the exact solutions of various nonlinear PDEs for some important contributions, refer to [11] [18]. In view of the algorithmic steps, we proceed as follows: Let us first consider the Lie group of point transformations t = t + ξ X, σ + O ɛ 2, θ = θ + τ X, σ + O ɛ 2, σ = σ + φ X, σ + O ɛ 2, 4 where X = t,θ, σ = P,Q and φ = η,φ, which leaves the system 2 invariant. In other words, the transformations are such that if σ is a solution of system 2 then σ is also a solution. The method for determining the symmetry group of system 2 mainly consists of finding the infinitesimals ξ, τ, η and φ, which are functions of t, θ, P and Q. We get the following relations from the coefficients of the first order of ɛ as η tt + tη t ξp t η θθ 2e 2P φ t Q t φ θ Q θ 2e 2P η Q 2 t Q 2 θ = 0, φ tt + φ t t ξq t φ θθ + 2φ t P t + 2η t Q t 2φ θ P θ 2η θ Q θ = 0, 5

3 3 Symmetry analysis and some solutions of Gowdy equations 17 where η t,η θ, φ t,φ θ, η tt, η θθ, φ θθ,φ θθ are extended prolonged infinitesimals acting on an enlarged space that includes all derivatives of the dependent variables P t, P θ, Q t, Q θ, P tt, P θθ, Q tt and Q θθ. The infinitesimals are determined from invariance conditions 5, by setting the coefficients of different differentials equal to zero. We obtain a large number of PDEs in ξ, τ, η and φ that need to be satisfied. The general solution of this large system provides the following forms for the infinitesimal elements ξ, τ, η and φ: τ = k 1 θ + k 2, ξ = k 1 t, η = k 3, φ = k 4 k 3 Q, where k i, i = 1,2,3,4 are arbitrary constants. The Lie algebra associated with the system of 2 consists of the following four vector fields: 6 V 1 = θ, V 2 = Q, V 3 = θ θ + t t, V 4 = P Q Q SYMMETRY REDUCTIONS AND SOLUTIONS One of the main purposes for calculating symmetries of a differential equation is to use them for obtaining symmetry reductions and finding exact solutions. In this section, we will make use of the linear combination of system of vector fields 7 and reduce the system equations 2 to systems of ODEs. The similarity variables and the similarity solutions of the Gowdy equation can be obtained by solving characteristic equation given dt ξ = dθ τ = dp η = dq φ. 8 The general solution of these equations involves three variables; one becomes the new independent variable ζ and the other two, F and G are dependent variables VECTOR FIELD V 3 For this vector field, on solving the system of equations 2 and 8 we obtain ζ = t θ,p = F ζ,q = Gζ. 9

4 18 Rajeev Kumar, R.K.Gupta, S.S.Bhatia 4 By using these similarity variables and similarity solutions in system 2, the reduced system of ODEs is as follows: F 1 + ζ 2ζ F ζ 2 F e 2F 1 ζ 2 G 2 = 0, G 1 + ζ 2ζ G ζ 2 G ζ 2 10 F G = 0. Solution of the system 10 of ODEs is as follow: Let Gζ = exp F ζh ζ, F ζ = N ζ. Using these substitutions our system reduce to N ζ + N 2Nζ 2 N ζ 3 ζn 2 H 2 + 2ζNHH H ζ N Hζ ζhn 2 NH + 2NHζ 2 + H ζh 2 + ζ 3 N 2 H 2 2ζ 3 NHH + ζ 3 H 2 = 0, 2ζ 2 H + ζ 3 N H + ζ 3 N 2 H ζ 3 H = 0. We arrive at following cases: Case 1: Hζ = 0, that is not a physically interesting case. Case 2: Hζ = ι, where ι represents the complex number iota. With this, our system 11 reduced to single equation: ζ 1 ζζ N + ζ 2 1 ζζ N ζ + 1 2ζ 3 4ζ N ζ 3 + 2ζ 2 + ζ 1 = 0, 12 which can be further solve to give solution ζ 2 1ζ + c 1 ln ζ + ζ 2 1 ζ 2 1ζ + c 1 ζ + c 2 N ζ = c 1 ln ζ + ζ ζ + 1, 13 ζ 2 1ζ where c 1, c 2 are constants. Using above we get final solution of system 2, ζ 2 1ζ + c 1 ln ζ + ζ 2 1 ζ 2 1ζ + c 1 ζ + c 2 P = c 1 ln ζ + ζ ζ + 1 dζ + d 1, 14 ζ 2 1ζ ζ 2 1ζ + c 1 ln ζ + ζ 2 1 ζ 2 1ζ + c 1 ζ + c 2 Q = ιexp c 1 ln ζ + ζ ζ + 1 dζ + d 1, ζ 2 1ζ where ζ = t θ and d 1 is constant. On substituting 14 in 3, we get λ =constant.

5 5 Symmetry analysis and some solutions of Gowdy equations VECTOR FIELD V 1 Following the same way as above, we get ζ = t,p = F ζ,q = Gζ. 15 Substituting 15 into system of equations 2 yields a reduced ODEs as: F + F ζ e2f G 2 = 0, G + G ζ + 2F G = 0. Solution of this system 16 of ODEs is as follows: Let Gζ = exp F ζh ζ, F ζ = N ζ. Using these substitutions our system reduce to N + N ζ N 2 H 2 + 2NHH H 2 = 0, ζh ζn H + H NH ζn 2 H = We arrive at following cases: Case 1: Hζ = 0, that is not a physically interesting case. Case 2: Hζ = ι, where ι represents the complex number iota. With this, our system 17 reduced to single equation: 1 ζ 2 N + N ζ 3ζN = 0, 18 which can be further solve to give solution: N ζ = ζ2 + d 2 ζζ 2 1, 19 where d 2 is constant. Using these results we get final solution of system 2 as follows: [ ] t P = log 2 1 ] d 2 + log[ t t 2 1, [ ] t Q = ιexp log 2 1 d 2 + log[ t t 2 1] 20. On substituting 20 in 3, we get λ =constant.

6 20 Rajeev Kumar, R.K.Gupta, S.S.Bhatia VECTOR FIELD V 1 + V 4 Corresponding to this vector field, the forms of the similarity variable and similarity solution are as follows: ζ = θ,p = F ζ,q = Gζ, 21 using above the system equations 2 reduces to the following ODEs: F e 2F G 2 = 0, G 2F G = 0. Solution of this system 22 of ODEs is as follows: Let Gζ = exp F ζh ζ, F ζ = N ζ. Using these substitutions our system reduce to N N 2 H 2 + 2NHH H 2 = 0, H N H 4NH + 3N 2 H = 0. We arrive at following cases: Case 1: Hζ = 0, that is not a physically interesting case. Case 2: Hζ = ι, where ι represents the complex number iota. With this, our system 23 reduced to single equation: ζn + N 2 ζ 2 2ζ + 4N + 3N 2 ζ + 1 = 0, 24 which can be further solve to give solution: N ζ = ζ d 3 d ζ + ζ ζ 1 2 5, 25 where d 3 is constant. Using these results we get final solution of system 2 as θ d 3 P = d θ + θ θ 1 dθ, 2 5 [ ] θ d 3 Q = ιexp d θ + θ θ 1 dθ. 2 5 On substituting 26 in 3, we get λ =constant CONCLUSION This paper intended to present some key aspects on how a vacuum Einstein field equations whose evolution is described by a nonlinear differential equation known as Gowdy equation, can be studied using the Lie classical symmetry method. The main

7 7 Symmetry analysis and some solutions of Gowdy equations 21 steps which have to be done in order to find a set of exact solutions are: i determination of the general form for the symmetry operator; ii based on the different set of operators and using the similarity reduction procedure, a complete set of invariant solutions can be generated; iii last but not least, a special method can be applied in order to find the solution. This is a new exact solution of the nonlinear system of vacuum Einstein field equations that are not found in the literature. REFERENCES 1. R.H. Gowdy, Ann. Phys. NY. 83, M. Tanimoto, J. Math. Phys. 39, S. Kichenassamy and A. Rendall, Class. Quantum Grav. 15, A.D. Rendall and M. Weaver, Class. Quantum Grav. 18, A. Einstein, Annal. Physik 17, A. Sáncheza, A. Maćyasa and H. Quevedo, Revista Mexicana De Física 53, A.D. Rendall, J. of Geo. and Phys. 62, S. Lie, Vorlesungenüber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen Teuber, Leipzig, 1891; reprinted by Chelsea, New York, P.J. Olver, Applications of Lie Groups to Differential Equations, New York, Springer Verlag, G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations, New York, Appl. Math. Sci., 154, Springer, K. Singh and R.K. Gupta, Int. J. of Eng. Sci., 44, R.K. Gupta and K. Singh, Comm. in Nonlinear Sci. and Num. Simu. 16, H. Triki, A. Yildirim, T. Hayat, O.M. Aldossary and A. Biswas, Rom. J. Phys. 57, A. Biswas, A. Yildirim, T. Hayat, O.M. Aldossary and R. Sassaman, Proc. of the Romanian Acad. A 13, G. Ebadi, N. Yousefzadeh Fard, A.H. Bhrawy, S. Kumar, H. Triki, A. Yildirim and A. Biswas, Rom. Rep. Phys. 65, A. Biswas, S. Kumar, E.V. Krishnan, B. Ahmed, A. Strong, S. Johnson and A. Yildirim, Rom. Rep. Phys. 65, G. Ebadi, A. Mojaver, H. Triki, A. Yildirim and A. Biswas, Rom. J. Phys. 58, N. Goyal and R.K. Gupta, Chin. Phys. B 20,